Theorems · Theorem · linear algebra

Submodule.iSup_induction

∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_8}
  (p : ι → Submodule R M) {motive : M → Prop} {x : M},
  x ∈ ⨆ i, p i →
    (∀ (i : ι), ∀ x ∈ p i, motive x) → motive 0 → (∀ (x y : M), motive x → motive y → motive (x + y)) → motive x

An induction principle for elements of ⨆ i, p i. If C holds for 0 and all elements of p i for all i, and is preserved under addition, then it holds for all elements of the supremum of p.

Defined in

Mathlib.LinearAlgebra.Span.Basic

Cited by

4 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModule